Introduction to the Variational Formulation in Mechanics

14Applications of Multiscale Modeling

14.1 Introduction

This chapter aims to present the application of the Method of Multiscale Virtual Power in two problems that appear in solid mechanics. For these two examples we will assume that the bodies are constituted by highly heterogeneous and stable materials at the microscale in the sense that, during the whole loading process, no strain localization, failure or damage phenomena, leading to loss of structural stability, is observed. In turn, we will consider the effects of dynamic forces (inertia) in the microscale in order to study the impact that these forces have in the definition of the macroscale model.

In the first example we consider both macroscale and microscale kinematics to be the same, and that we are within the range of finite deformations. The second example differs in the sense that different types of constitutive models are considered at both scales. In particular, we consider a given constitutive material response at the macroscale, but at the same time the continuum is considered to behave as incompressible. At the microscale, the incompressibility constraint will not be considered. These examples are sufficient to show the versatility of the proposed method when approaching different situations usually encountered in multiscale modeling. In this sense, reading the scientific contributions published by the authors, where we report the application of this technique in diverse problems including, among others, fracture ...